Journal of Difference Equations

Volume 2015, Article ID 549390, 21 pages

http://dx.doi.org/10.1155/2015/549390

## Stability of Hyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

^{1}School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia^{2}Department of Basic Sciences, King Saud bin Abdulaziz University for Health Sciences, Riyadh 11426, Saudi Arabia

Received 4 January 2015; Accepted 25 March 2015

Academic Editor: Richard Saurel

Copyright © 2015 S. Atawna et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Our goal in this paper is to investigate the global asymptotic stability of the hyperbolic equilibrium solution of the second order rational difference equation , , where the parameters and , , , , are positive real numbers and the initial conditions are nonnegative real numbers such that . In particular, we solve Conjecture 5.201.1 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.2 in Kulenović and Ladas monograph (2002).

#### 1. Introduction

Rational difference equations, particularly bilinear ones, that is,attracted the attention of many researchers recently. For example, see the articles [1–9]. As it turns out, many models, such as population models in mathematical biology, are members of the family of rational difference equations. The behavior of solutions of rational difference equations can provide prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one. Hence, the study of this family of equations is important from both a theoretical point of view and the point of view of applications.

For the general theory of difference equations, one can refer to Agarwal [10], Elaydi [11], Kelley and peterson [12], and the monograph of Kocic and Ladas [13]. Many rational difference equations were studied extensively in [14, 15] and the references cited therein.

Asystematic study of the second order rational difference equation of the formwhere the parameters , , , , , and the initial conditions , are nonnegative real numbers, was considered in the Kulenović and Ladas monograph [15]. They came up with the idea of setting one or more parameters in (2) to zero and studying the resulting equation with fewer parameters. This approach gives rise to 49 different cases which exhibit variety dynamics. They presented the known results such as [16–23]. Next, Kulenović and Ladas [15] derived several ones on the boundedness, the global stability, and the periodicity of solutions of all rational difference equations of the form (2). Furthermore, they posed several open problems and conjectures related to this equation and its functional generalization.

Even after a sustained effort by many researchers such as [24–28], there were some cases of the 49 different cases that have not been investigated till 2007. Amleh et al. in [29, 30] give an up-to-date account on recent developments related to (2) up to 2007. Furthermore, they reposed several open problems and conjectures related to this equation.

Camouzis and Ladas in [14] summarize the progress up to 2008 in the study of the 49 cases of (2) as subcases of the 255 special cases of the more general third-order rational difference equationwhere the parameters , , , , , , , and the initial conditions , , are nonnegative real numbers. In their book, Camouzis and Ladas [14] have posed a series of open problems and conjectures related to (3). In addition, they reposed open problems and conjectures on these remaining equations of (2) that have resisted analysis so far.

Recently, the work done by many researchers such as [31–39] have solved many open problems and conjectures proposed in [14, 15, 29, 30] related to (2) and have led to the development of some general theory about difference equation. However, as confirmed by Professor Kulenović (personal communication, August 24, 2014), the case remains open.

Our approach handles the aforementioned case as well as other cases. Furthermore, the results of this paper improve and extend the asymptotic results in [15, Chapter 11]. Indeed, our results provide affirmative answer to the following conjecture proposed by Camouzis and Ladas in [14, Conjecture 5.201.1].

Conjecture 1. *This shows that, for the equilibrium of (2), *

It is worth mentioning that the aforementioned conjecture appeared previously in the Kulenović and Ladas monograph [15, Conjecture 11.4.2].

To this end and using the transformation equation (2) reduces towhere are positive real numbers and the initial conditions , are nonnegative real numbers.

The periodic character of positive solutions of (6) has been investigated by the authors in [40]. They showed that the period-two solution is locally asymptotically stable if it exists.

Our results, together with the established results in [15, 40], give a complete picture of the nature of solutions of the second order rational difference equation of the form (2). We believe that our results are important stepping stone in understanding the behavior of solutions of rational difference equations which provides prototypes towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of higher order.

That being said, the remainder of this paper is organized as follows. In the next section, a brief description of some definitions and results from the literature that are needed to prove the main results in this paper is given. It is worth mentioning that there are few global attractivity results in the literature that can be applied to rational difference equations of the form (2). Next we establish our main results in Sections 3–5. We determine the local stability character of (2) in Section 3. Section 4 examines the existence of intervals which attract all solutions of (2). In Section 5, we investigate the global asymptotic stability of the hyperbolic equilibrium solution of (6). Next, in Section 6 we consider several numerical examples generated by MATLAB to illustrate the results of the previous sections and to support our theoretical discussion. Finally, we conclude in Section 7 with suggestion for future research.

#### 2. Preliminaries

For the sake of self-containment and convenience, we recall the following definitions and results from [15].

Let be a nondegenerate interval of real numbers and let be a continuously differentiable function. Then for every set of initial conditions , the difference equationhas a unique solution .

A constant sequence, for all where , is called an* equilibrium solution* of (8) if

*Definition 2. *Let be an equilibrium solution of (8). (i) is called* locally stable* if, for every , there exists such that, for all , with , we have (ii) is called* locally asymptotically stable* if it is locally stable and if there exists , such that, for all , with , we have (iii) is called a* global attractor* if, for every , we have (iv) is called* globally asymptotically stable* if it is locally stable and a global attractor.(v) is called* unstable* if it is not stable.(vi) is called a* source*, or a* repeller*, if there exists such that, for all , with , there exists such that Clearly a source is an unstable equilibrium.

*Definition 3. *Let denote the partial derivatives of evaluated at the equilibrium of (8). Then the equationis called the* linearized equation associated with (8)* about the equilibrium solution .

Theorem 4 (linearized stability). *(a) If both roots of the quadratic equation**lie in the open unit disk , then the equilibrium of (8) is locally asymptotically stable.**(b) If at least one of the roots of (16) has absolute value greater than one, then the equilibrium of (8) is unstable.**(c) A necessary and sufficient condition for both roots of (16) to lie in the open unit disk is**In this case the locally asymptotically stable equilibrium is also called a sink.**(d) A necessary and sufficient condition for both roots of (16) to have absolute value greater than one is **In this case is a repeller.**(e) A necessary and sufficient condition for one root of (16) to have absolute value greater than one and for the other to have absolute value less than one is**In this case the unstable equilibrium is called a saddle point.**(f) A necessary and sufficient condition for a root of (16) to have absolute value equal to one is**or**In this case the equilibrium is called a nonhyperbolic point.*

Theorem 5. *Consider the difference equation (8). Let be some interval of real numbers and assume that**is a continuous function satisfying the following properties. *(a)* is nonincreasing in for each , and is nondecreasing in for each ;*(b)*the difference equation (8) has no solutions of prime period two in ;**then (8) has a unique equilibrium and every solution of (8) converges to .*

Theorem 6. *Let be an interval of real numbers and assume that**is a continuous function satisfying the following properties. *(a)* is nondecreasing in for each , and is nonincreasing in for each ;*(b)*if is a solution of the system* *then .**Then (8) has a unique equilibrium and every solution of (8) converges to .*

Theorem 7. *Let be an interval of real numbers and assume that**is a continuous function satisfying the following properties. *(a)* is nonincreasing in each of its arguments;*(b)*if is a solution of the system* *then .**Then (8) has a unique equilibrium and every solution of (8) converges to .*

Theorem 8. *Let be an interval of real numbers and assume that**is a continuous function satisfying the following properties. *(a)* is nondecreasing in each of its arguments;*(b)*the equation* *has a unique positive solution.**Then (8) has a unique equilibrium and every solution of (8) converges to .*

The following result from [40] will be useful in the sequel.

Theorem 9. *(a) When **equation (6) has no nonnegative prime period-two solution.**(b) When **equation (6) has prime period-two solution, **if and only if condition**where and are the positive and distinct solutions of the quadratic equation *

*3. Local Stability*

*In this section, we address the local stability of the equilibrium of (6) when all parameters are positive. In particular, we give explicit conditions on the parameter values of (6) for the equilibrium to be locally asymptotically stable.*

*Equation (6) has a unique positive equilibrium given byThe linearized equation associated with (6) about the equilibrium solution is given byTherefore, its characteristic equation isBy applying linearized stability (Theorem 4(c)) we have the following result.*

*Theorem 10. (a) Assume thatthen the positive equilibrium of (6) is locally asymptotically stable.*

(b) Assume thatthen the positive equilibrium of (6) is locally asymptotically stable if and only if

*Proof. *By employing linearized stability (Theorem 4(c)) we see that condition (17) is equivalent to the following three inequalities:Inequality (40) is satisfied if and only if , which is always satisfied by (34). Inequality (42) is satisfied if and only if , which is always satisfied since , , are positive.

Inequality (41) is satisfied if and only if . Hence is locally asymptotically stable (sink) if and only ifClearly the equilibrium is the positive solution of the quadratic equation Now set then Inequality (43) is satisfied if and only if either or that is, from which (39) follows.

The proof is complete.

*4. Invariant Intervals*

*4. Invariant Intervals*

*In this section, we investigate the invariant intervals for (6) in order to obtain convergence results for the solutions of (6).*

*Let be a positive solution of (6). Then we have the following identities:We consider the cases where , , and .*

*4.1. Case 1: *

*4.1. Case 1:*

*When , Identities (49) and (50) are equivalent to the following:The following remark is straightforward from Identities (51) and (52).*

*Remark 11. *Let be a positive solution of (6). Then the following statements are true when . (1) if and only if .(2) if and only if .

*Table 1 gives the signs of and of (6) in all possible nondegenerate cases when .*